3.42 \(\int \frac {(a+b \tanh ^{-1}(c+d x))^2}{e+f x} \, dx\)

Optimal. Leaf size=214 \[ -\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \text {Li}_2\left (1-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{f}-\frac {b^2 \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e-c f+f) (c+d x+1)}\right )}{2 f}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{c+d x+1}\right )}{2 f} \]

[Out]

-(a+b*arctanh(d*x+c))^2*ln(2/(d*x+c+1))/f+(a+b*arctanh(d*x+c))^2*ln(2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f+b*(a
+b*arctanh(d*x+c))*polylog(2,1-2/(d*x+c+1))/f-b*(a+b*arctanh(d*x+c))*polylog(2,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x
+c+1))/f+1/2*b^2*polylog(3,1-2/(d*x+c+1))/f-1/2*b^2*polylog(3,1-2*d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/f

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Rubi [A]  time = 0.15, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6111, 5922} \[ -\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}+\frac {b \text {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{f}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{2 f}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{c+d x+1}\right )}{2 f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(c+d x+1) (-c f+d e+f)}\right )}{f}-\frac {\log \left (\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c + d*x])^2/(e + f*x),x]

[Out]

-(((a + b*ArcTanh[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e
+ f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*Ar
cTanh[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1
+ c + d*x)])/(2*f) - (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)

Rule 5922

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^2*Log[
2/(1 + c*x)])/e, x] + (Simp[((a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(
b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e, x] - Simp[(b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(
d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(b^2*PolyLog
[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2,
0]

Rule 6111

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{e+f x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{\frac {d e-c f}{d}+\frac {f x}{d}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+c+d x}\right )}{f}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{f}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b^2 \text {Li}_3\left (1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{2 f}\\ \end {align*}

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Mathematica [C]  time = 24.28, size = 1757, normalized size = 8.21 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTanh[c + d*x])^2/(e + f*x),x]

[Out]

(a^2*Log[e + f*x])/f - ((2*I)*a*b*(I*ArcTanh[c + d*x]*(-Log[1/Sqrt[1 - (c + d*x)^2]] + Log[I*Sinh[ArcTanh[(d*e
- c*f)/f] + ArcTanh[c + d*x]]]) + ((-I)*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])^2 - (I/4)*(Pi - (2*I)
*ArcTanh[c + d*x])^2 + 2*(I*ArcTanh[(d*e - c*f)/f] + I*ArcTanh[c + d*x])*Log[1 - E^((2*I)*(I*ArcTanh[(d*e - c*
f)/f] + I*ArcTanh[c + d*x]))] + (Pi - (2*I)*ArcTanh[c + d*x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))] - (
Pi - (2*I)*ArcTanh[c + d*x])*Log[2*Sin[(Pi - (2*I)*ArcTanh[c + d*x])/2]] - 2*(I*ArcTanh[(d*e - c*f)/f] + I*Arc
Tanh[c + d*x])*Log[(2*I)*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] - I*PolyLog[2, E^((2*I)*(I*ArcTanh[(
d*e - c*f)/f] + I*ArcTanh[c + d*x]))] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTanh[c + d*x]))])/2))/f + (b^2*(d*e -
c*f + f*(c + d*x))*((2*ArcTanh[c + d*x]^2*(d*e*ArcTanh[c + d*x] - (1 + c)*f*ArcTanh[c + d*x] + 3*(d*e - c*f)*
Log[1 + E^(-2*ArcTanh[c + d*x])]) + (-6*d*e*ArcTanh[c + d*x] + 6*c*f*ArcTanh[c + d*x])*PolyLog[2, -E^(-2*ArcTa
nh[c + d*x])] + (-3*d*e + 3*c*f)*PolyLog[3, -E^(-2*ArcTanh[c + d*x])])/(6*f*(-(d*e) + c*f)) - ((-(d*e) - f + c
*f)*(-(d*e) + f + c*f)*(-3*d*e*ArcTanh[c + d*x]^3 + f*ArcTanh[c + d*x]^3 + 3*c*f*ArcTanh[c + d*x]^3 - (2*Sqrt[
1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]*f*ArcTanh[c + d*x]^3)/E^ArcTanh[(d*e - c*f)/f] - (3*I)*d*e*Pi*ArcTanh[c
+ d*x]*Log[(1 + E^(2*ArcTanh[c + d*x]))/(2*E^ArcTanh[c + d*x])] + (3*I)*c*f*Pi*ArcTanh[c + d*x]*Log[(1 + E^(2
*ArcTanh[c + d*x]))/(2*E^ArcTanh[c + d*x])] + 3*d*e*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + Arc
Tanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[1 - E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 3*d*e*ArcT
anh[c + d*x]^2*Log[1 + E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*Log[1 + E^(Ar
cTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*d*e*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[(I/2)*E^(-ArcTan
h[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])))] - 6*c*f*ArcTanh
[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[(I/2)*E^(-ArcTanh[(d*e - c*f)/f] - ArcTanh[c + d*x])*(-1 + E^(2*(ArcTanh[
(d*e - c*f)/f] + ArcTanh[c + d*x])))] + 3*d*e*ArcTanh[c + d*x]^2*Log[(d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 +
c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^ArcTanh[c + d*x])] - 3*c*f*ArcTanh[c + d*x]^2*L
og[(d*e*(1 + E^(2*ArcTanh[c + d*x])) - (1 + c - E^(2*ArcTanh[c + d*x]) + c*E^(2*ArcTanh[c + d*x]))*f)/(2*E^Arc
Tanh[c + d*x])] + (3*I)*d*e*Pi*ArcTanh[c + d*x]*Log[1/Sqrt[1 - (c + d*x)^2]] - (3*I)*c*f*Pi*ArcTanh[c + d*x]*L
og[1/Sqrt[1 - (c + d*x)^2]] - 3*d*e*ArcTanh[c + d*x]^2*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d
*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] + 3*c*f*ArcTanh[c + d*x]^2*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*
f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]] - 6*d*e*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x
]*Log[I*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] + 6*c*f*ArcTanh[(d*e - c*f)/f]*ArcTanh[c + d*x]*Log[I
*Sinh[ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x]]] + 6*(d*e - c*f)*ArcTanh[c + d*x]*PolyLog[2, -E^(ArcTanh[(d*e
- c*f)/f] + ArcTanh[c + d*x])] + 6*(d*e - c*f)*ArcTanh[c + d*x]*PolyLog[2, E^(ArcTanh[(d*e - c*f)/f] + ArcTan
h[c + d*x])] - 6*d*e*PolyLog[3, -E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*c*f*PolyLog[3, -E^(ArcTanh
[(d*e - c*f)/f] + ArcTanh[c + d*x])] - 6*d*e*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])] + 6*c*f
*PolyLog[3, E^(ArcTanh[(d*e - c*f)/f] + ArcTanh[c + d*x])]))/(3*f*(-(d*e) + c*f)*(d^2*e^2 - 2*c*d*e*f + (-1 +
c^2)*f^2))))/(d*(e + f*x))

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {artanh}\left (d x + c\right ) + a^{2}}{f x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(d*x+c))^2/(f*x+e),x, algorithm="fricas")

[Out]

integral((b^2*arctanh(d*x + c)^2 + 2*a*b*arctanh(d*x + c) + a^2)/(f*x + e), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(d*x+c))^2/(f*x+e),x, algorithm="giac")

[Out]

integrate((b*arctanh(d*x + c) + a)^2/(f*x + e), x)

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maple [C]  time = 0.97, size = 1984, normalized size = 9.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(d*x+c))^2/(f*x+e),x)

[Out]

a*b/f*ln((d*x+c)*f-c*f+d*e)*ln(((d*x+c)*f-f)/(c*f-d*e-f))-a*b/f*ln((d*x+c)*f-c*f+d*e)*ln(((d*x+c)*f+f)/(c*f-d*
e+f))+b^2*c/(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+b^2*c/(c*f-d
*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-I*b^2/f*Pi*arctanh(d*x+c)^2
+1/2*I*b^2/f*arctanh(d*x+c)^2*Pi*csgn(I/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))*csgn(I*(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^2
))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f))*csgn(I*(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^
2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))+b^2*ln(
(d*x+c)*f-c*f+d*e)/f*arctanh(d*x+c)^2-b^2/f*arctanh(d*x+c)^2*ln(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^
2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)-b^2/f*arctanh(d*x+c)*polylog(2,-(d*x+c+1)^2/(1-(d*x+c)
^2))-b^2/(c*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+a*b/f*dilog(
((d*x+c)*f-f)/(c*f-d*e-f))-b^2/(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1-(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d
*e-f))-a*b/f*dilog(((d*x+c)*f+f)/(c*f-d*e+f))-1/2*b^2*c/(c*f-d*e-f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+
c)^2)/(-c*f+d*e-f))+2*a*b*ln((d*x+c)*f-c*f+d*e)/f*arctanh(d*x+c)+1/2*d*b^2/f*e/(c*f-d*e-f)*polylog(3,(c*f-d*e-
f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+I*b^2/f*arctanh(d*x+c)^2*Pi*csgn(I*(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^2
))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))^2-1/2*I*
b^2/f*arctanh(d*x+c)^2*Pi*csgn(I*(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x
+c+1)^2/(1-(d*x+c)^2))*f)/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))^3-1/2*I*b^2/f*arctanh(d*x+c)^2*Pi*csgn(I/(1+(d*x+c+1)
^2/(1-(d*x+c)^2)))*csgn(I*(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2
/(1-(d*x+c)^2))*f)/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))^2-d*b^2/f*e/(c*f-d*e-f)*arctanh(d*x+c)^2*ln(1-(c*f-d*e-f)*(d
*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))-d*b^2/f*e/(c*f-d*e-f)*arctanh(d*x+c)*polylog(2,(c*f-d*e-f)*(d*x+c+1)^2/(
1-(d*x+c)^2)/(-c*f+d*e-f))-1/2*I*b^2/f*arctanh(d*x+c)^2*Pi*csgn(I*(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c+
1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f))*csgn(I*(c*f*(1+(d*x+c+1)^2/(1-(d*x+c)^2))+(-(d*x+c
+1)^2/(1-(d*x+c)^2)-1)*e*d+(1-(d*x+c+1)^2/(1-(d*x+c)^2))*f)/(1+(d*x+c+1)^2/(1-(d*x+c)^2)))^2+1/2*b^2/(c*f-d*e-
f)*polylog(3,(c*f-d*e-f)*(d*x+c+1)^2/(1-(d*x+c)^2)/(-c*f+d*e-f))+a^2*ln((d*x+c)*f-c*f+d*e)/f+1/2*b^2/f*polylog
(3,-(d*x+c+1)^2/(1-(d*x+c)^2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \log \left (f x + e\right )}{f} + \int \frac {b^{2} {\left (\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )\right )}^{2}}{4 \, {\left (f x + e\right )}} + \frac {a b {\left (\log \left (d x + c + 1\right ) - \log \left (-d x - c + 1\right )\right )}}{f x + e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(d*x+c))^2/(f*x+e),x, algorithm="maxima")

[Out]

a^2*log(f*x + e)/f + integrate(1/4*b^2*(log(d*x + c + 1) - log(-d*x - c + 1))^2/(f*x + e) + a*b*(log(d*x + c +
1) - log(-d*x - c + 1))/(f*x + e), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{e+f\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c + d*x))^2/(e + f*x),x)

[Out]

int((a + b*atanh(c + d*x))^2/(e + f*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{2}}{e + f x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(d*x+c))**2/(f*x+e),x)

[Out]

Integral((a + b*atanh(c + d*x))**2/(e + f*x), x)

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